I would like to 'draw' a basic 2D mesh of any shape using my mouse
I would argue that this approach doesn't really save too much time, since handdrawn shapes must be cleaned... manually. But I concede that it's probably suitable if you're just making a sketch!

Anyway: you can definitely do it using grease pencil! What is the problem you're experiencing?

This is a possible workflow. Precondition: use Blender 2.80 or newer.


Add > Grease Pencil > Blank
Get into an orthographic view (e.g. Front Ortho)
Entrer "Draw Mode", draw; then go back to "Object Mode"
Right click on your object > Convert to Bezier Curve
In the Outliner, hide the grease pencil object and select the newly created Bezier curve.

In my version, I have a little bug whereas the Bezier is not immediately visible. Hop in and out of Edit Mode (Tab) to make it visible


Your curve has probably too many handles. To reduce them, enter "Edit Mode", select all the handles (A), then do Right Click > Decimate Curve. In the box that appears, select a suitable value.


To avoid your mesh to have still too many handles, go in the Curve settings and change Resolution Preview U to 2 or 3
Right Click > Convert to Mesh
You've got your mesh! You can use it as any other mesh. For instance, you can use the Screw modifier